Database Reference
In-Depth Information
Let us show that i 1 is monic and epic as well. For any two morphisms g 1 ,g 2 :
1
1
C such that i 1
g 1 =
i 1
g 2 , that is, i 1
g 1 =
g 1 =
=
D
id C ,
g 1 ,
i 1
g 1
i 1
g 2
1
i 1 g 2 = g 2 ,
and by point
3 of Definition 23 ) and hence i 1 is a monomorphism. For any two g 1 =[
, we obtain g 1 = g 2 (from
{ g 1 }=
=
={ g 2 }
1
k 1 ,
]
and
1
0
1
g 2 =[
]:
+⊥
D such that g 1
i 1 =
g 2
g 1 , that is, g 1
i 1 =[
]◦
k 2 ,
C
k 1 ,
1
1
1
id C ,
= k 1 ,
= k 1 (component
is eliminated according to point 4 in
1
1
1
Definition 20 )
= g 2 i 1 =[ k 2 ,
]◦
id C ,
= k 2 ,
= k 2 , we obtain k 1 = k 2 ,
thus g 1 =
g 2 and hence i 1 is an epimorphism.
0 is isomorphic to C and that two isomorphic
objects in a category can be substituted, we can eliminate for complex objects all
their components which are equal to the empty database
Remark From the fact that C
0 and, generally, consider
only the strictly-complex objects. Now we obtain the first fundamental property of
DB category which confirms that it is fundamentally different from Set :
Proposition 5
The morphisms
[
id C , id C ]:
C
C
C are epic and monic but not
isomorphic . Consequently , DB is not a topos .
Proof Let for any two morphisms g 1 =
h 1 ,k 1 :
C
C
C and g 2 =
h 2 ,k 2 :
C
C
C ,
[
id C , id C ]◦
g 1 =[
id C , id C ]◦
g 2 :
C
C . Let us show that g 1 =
g 2 .
In fact,
[
id C , id C ]◦
g 1 =[
id C , id C ]◦
h 1 ,k 1 =
id C
h 1 , id C
k 1 =
h 1 ,k 1 :
C
C and
[
id C , id C ]◦
g 2 =
h 2 ,k 2 :
C
C . Consequently,
h 1 ,k 1 =
h 2 ,k 2
=
=
and, from Definition 23 , there exists a bijection σ such that σ(h 1 )
h 2 , σ(k 1 )
k 2 with h 1 = σ(h 1 )
= h 2 and k 1 = σ(k 1 )
= k 2 , so that h 1 =
h 2 and k 1 =
k 2 and
hence g 1 =
h 1 ,k 1 =
h 2 ,k 2 =
g 2 . Consequently,
[
id C , id C ]:
C
C
C is a
monomorphism.
Let for any two morphisms g 1 :
C and g 2 :
C , g 1 ◦[
id C , id C ]=
C
C
1
g 2 ◦[
id C , id C ]
(the case when g 1 = g 2 =⊥
is trivial and hence we consider the
1 ). Let us show that g 1 = g 2 . In fact,
cases when g 1 and g 2 are different from
g 1 ◦[
id C , id C ]=[
g 1 ,g 1 ]:
C
C
C and
[
g 2
id C , id C ]=[
g 2 ,g 2 ]:
C
C .
Consequently,
[
g 1 ,g 1 ]=[
g 2 ,g 2 ]
and, from point 3 of Definition 23 , g 1 =
g 2 .
Therefore,
[
id C , id C ]:
C
C
C is an epimorphism. Let us show that f
=
[
id C , id C ]:
C
C
C is not an isomorphism: If we suppose that it is an iso-
f 1
morphism then f
:
C
C has to be the identity arrow id C .Fromthe
fact that f 1
OP
id OP
C
, id O C =
=[
id C , id C ]
=
id C , id C :
C
C
C , so that
f 1
=[
id C , id C ]◦
id C , id C =
id C
id C , id C
id C =
id C , id C =
f
id C (by
point 2 of Definition 23 ), i.e., the properties of an isomorphism are not satisfied.
From the fact that in any topos category if a morphism is both epic and monic
then it must also be an isomorphism, we conclude that DB is not a topos. More about
this will be considered in Sect. 9.1 dedicated to topological properties of DB .
Thus, DB is not a topos. However, there is a subset of morphisms for which the
topological properties hold so that if a morphism is both epic and monic then it
must also be an isomorphism: it is the set of all simple arrows, as we will show in
the next corollary, so that the first step is to analyze the topological properties of the
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